Fact: The correct name is tzadi, with no “k” sound at the end. However, the use of tzaddik as a viable alternative has gained some acceptance.
Background:The letter is referred to as tzadi in the Talmud. In the Yerushalmi (Megillah 1:9), tzadi (spelled tzadi daled yud) is listed along with other letters that have a final form. Similarly, the Bavli (Shabbat 104a) provides a midrash on the names of the letters of the alphabet in which it spells out the name tzadi. On the previous page, the Talmud, when listing letters that can be erroneously interchanged when written, uses the word tzadin when referring to the plural of tzadi, providing further evidence that the name does not end with a kuf. The Talmud (Yoma 73b; cf. Yerushalmi Yoma 7:3) once again mentions the tzadi when questioning where the letter was located on the high priest’s breastplate (urim vetumim).
One of the earliest and most important Jewish philologists, Yona ibn Janach, mentions the word tzadi many times in his Sefer HaShorashim.
More recent evidence can also be found. The Sdei Chemed is arranged alphabetically and includes a section called “tzadi.” In the sections detailing the halachot of writing a Sefer Torah in the Beit Yosef (OC 36), Mishnah Berurah (OC 36, p. 64) and Aruch Hashulchan (OC 32:49, 36:22 and Yoreh Deah 274:10), tzadi is written out several times. Rav Hershel Schachter in MiPninei HaRav (2001, 267) also refers to the tzadi. The Encyclopedia Judaica (14:622) uses the term “sade.” Moreover, parallel letters in Arabic (sad), Syriac (sade) and Ethiopic (sadai), all closely related languages, have no “k” sound at the end.
Despite this, one still hears the letter referred to as tzaddik. The most likely reason for this is that since the kuf directly follows the tzadi in the Hebrew alphabet, many people inadvertently link the two. Another reason is that in Hebrew acrostics, the word tzaddik (as in righteous person) is frequently used for the letter tzadi, and in various midrashim, the word tzaddik appears in connection with the tzadi. Examples of the former include Psalms 145:17—which is included in Ashrei and, as such, known by heart by many—as well as Psalms 119:137 and 112:9. An example of the latter includes the midrash cited in the Talmudic statement above (Shabbat 104a).
Despite the error, the “letter” tzaddik is found in traditional sources. However, unlike the misconception regarding Cheshvan and Mar Cheshvan, this doesn’t seem to have any halachic ramifications. Consequently, it persists with little, if any, protest.
The earliest source that refers to the letter in question as tzaddik appears to be Sifrei  to Deuteronomy 6:9 where a scribe is cautioned not to mix up the letters tzaddik and gimmel. One might deduce from this that alternate pronunciation already existed in the fifth century (which is when the Sifrei was first written). However this is not the case. The first printed edition of the Sifrei (Venice, 1545) used the term tzaddik, however, earlier manuscripts all refer to the tzadi (according to the critical edition of the Sifrei by L. Finkelstein). Thus, it appears that the error may have been introduced by the printer. (Many false traditions have become well established because of uneducated printers.)
In the introduction to the Zohar, (2b) God is quoted as saying to the tzadi: “O tzadi, you are tzadi and you are tzaddik….” This is understood to mean that God was telling the tzadi that he is righteous (tzaddik). But it can be (mis)understood to mean that God was saying that the name of the letter is both tzadi and tzaddik. Rashi to Menachot 29b (s.v. Shatnez gaetz) calls the letter tzaddik  as well. In modern times the use of the term tzaddik has become even more popular. In Rachamim LeChaim, a commentary on the Rashba, the author, Rabbi Chaim Palache,  references another work of his, Kuntres Derech Yam, and refers the reader to section “tzaddik.” Rav Yitzchak Elchanan Spector misquotes the Talmud and calls the letter tzaddik. In some modern sources both options are offered, such as in the classic Hebrew/English dictionary by Reuben Alcalay (1990, p. 2143). The Modern English-Yiddish/Yiddish-English Dictionary by Uriel Weinreich lists the letter as tzaddik as well. Additional research is required to find the earliest use of the term tzaddik  in lieu of tzadi.
A Bar-Ilan CD-Rom search reveals hundreds of uses of tzadi in sources including the Bavli, Yerushalmi, Midrash Rabbah, Pirkei D’Rabbi Eliezer, Zohar, Zohar Chadash, Geonim, Rishonim and Acharonim on Tanach and Shas, codes and numerous responsa. Surprisingly, there are also tens of instances of the use of tzaddik to mean tzadi. For example, the Ramban refers to tzaddik (Exodus 25:29); however he also refers to tzadi (Exodus 16:1). Again, it is not clear whether a printer is at fault here. There are very few early uses of the term tzaddik, fewer than two score. However, several dozen can be found in nineteenth and twentieth century responsa and codes and thus cannot be ascribed to printers’ errors. For example Rav Eliezer Waldenberg uses both terms several times. Most contemporary authorities, however, are careful; Rav Ovadia Yosef mentions tzadi several dozen times and tzaddik only once when making a play on the name. Thus, although tzaddik has become an acceptable alternative and can be found in sources such as the Nodah B’Yehudah, Chatam Sofer, Mishnah Berurah and Kol Mevaser, it is clear that the correct name was and remains tzadi.
It is worth noting that the Yemenites have a slightly different custom regarding the names of the Hebrew letters. Rav Shlomo ben Rav Yechyei Amram Korach, one of the leading contemporary Yemenite authorities, brings down this custom in his work, Arichat Shulchan, (vol. 3, p. 166-168). Instead of referring to the end letters as sofit, the Yemenites refer to those letters with two forms as either peshutah (straight) or kefufah (bent), which is how the Talmud refers to them. Rav Shlomo refers to the letter under discussion as a tzad peshuta and tzad kefufah and emphasizes that it is neither tzadi nor tzaddik, but “tzad.”
The tzadi/tzaddik distinction has been put to good use. As an example of how a gadol can relate to children, the story is told that Rav Yaakov Kamenetsky, zt”l, would show a little boy the letter under discussion and have an earnest conversation with him about whether it is a tzaddik or a tzadi. 
1. Lived in the first half of the eleventh century. See Encyclopedia Judaica 8:1181-1186.
2. It should be noted that there are three places where the Mishnah Berurah uses the term tzaddik but twenty-seven where it uses the term tzadi.
3. See the chart in Peter T. Daniels, “Scripts of Semitic Languages” in Robert Hetzron’s The Semitic Languages (London, 1997), 35. I thank Dr. Adina Moshavi for this source.
4. See “Legal-ease: What’s the Truth about…Mar Cheshvan?” Jewish Action, fall 2000, 28-29.
5. Piska 36 in Va’etchanan, Finkelstein edition 5753, 65.
6. It is certainly possible that in Rashi, as in the Sifrei, the original was tzadi and a printer who thought it should be tzaddik changed it. Old manuscripts would have to be consulted to verify this.
7. She’eilot U’Teshuvot HaRashba, chelek 7, siman 343. Rashba was Rav Shlomo ben Aderet, 1235-1310.
8. Sephardic halachic authority (b. Izmir, 1788-1869). On Palache see Encyclopedia Judaica 13:17-18. Rachamin LeChaim can be found in the back of his book, Nishmat Kol Chai, vol. 2, as well as in some editions of the She’eilot U’Teshuvot HaRashba.
9. 1817-1896; She’eilot U’Teshuvot Ein Yitzchak, chelek aleph, Yoreh Deah, siman 28, par., 5, 6.
10. For a philosophical discussion of tzadi vs. tzaddik, see Eliyahu (Elias) Lipiner, Ideologye Fun Yiddish Alef-Beys (Buenos Aires, 1967) 165, 497-502.
11. This is preferred to the term sofit since there is no kof rishonit, for example.
12. The other letters that he gives slightly different names to are: be (bet), dal (dalet), va (vav), zan (zayin), yod (pronounced as such and not yud ), ahn (ayin—the ayin is only mentioned in a footnote; the author seems to have left the letter out in the text, which I assume he did accidentally), peh (pay) and kof (kuf ). See also Rav Yosef Kafich, Halichot Taiman, (5723), 50 for the names of the letters.
13. Yonason Rosenblum, Reb Yaakov: The Life and Times of Hagaon Rabbi Yaakov Kamenetsky (New York, 1993), 244. This book is based on the research of Rabbi Noson Kamenetsky. I thank Rabbi Aharon Marcus for pointing this story out to me.
Reprinted from JEWISH ACTION Magazine, Spring 5763/2003 issue